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The posterior ethmoidal nerve is a branch of the nasociliary nerve supplying sensory pathways from posterior ethmoidal and nearby sinus regions. Its distribution differs from the more anterior territory of the anterior ethmoidal nerve.

The perpendicular distance from a muscle force's line of action to the relevant axis of an anatomical joint. It can change with posture; multiplying it by the muscle force gives the corresponding torque.

For independent standard normal random variables $X,Y$, put $X=R\cos \Theta$, $Y=R\sin \Theta$, with $R\ge 0$ and $0\le \Theta <2\pi$. The polar coordinates Jacobian determinant is $r$, so the joint probability density is $r{e}^{-r^2/2}/(2\pi )$ on $r>0$. It factorizes into a Rayleigh distribution density for $R$ and a continuous uniform distribution density for $\Theta$. Consequently the radius and angle are independent. The change of variables for probability densities $S=R^2$ gives $f_S(s)=e^{-s/2}/2$ for $s>0$, the exponential distribution of rate $1/2$, also the chi-squared distribution with two degrees of freedom. Measurable transformation preserves independence, giving $\Theta \perp S$. The angle at $R=0$ is arbitrary on an event of probability zero.

Thus a sector with angular length $\ell$ has probability $\ell /(2\pi )$, independently of any radial restriction. For instance $Y>|X|$ occupies a sector of angle $\pi /2$ and has probability $1/4$.

The function mapping each possible element to its multiset multiplicity. Finite multisets correspond to nonnegative functions with finite support of a multiset. A functional representation can answer queries without supporting enumeration of all elements.

The number of times a value occurs in observed data. Keeping 26 indexed counters for lowercase alphabet characters gives constant-time frequency queries independent of document length.

The morphism ${\mathbb{A}}^1\to V(x^4-y^3)$ given by $t\mapsto (t^3,t^4)$ is bijective on points over an algebraically closed field, but is not an isomorphism. At a nonzero curve point, $t=y/x$ is the unique inverse parameter, while the origin has unique preimage zero. The induced map on coordinate rings has image $k[t^3,t^4]\subset k[t]$, which lacks the monomial $t$: all its monomial exponents have the form $3a+4b$, with $a,b\ge 0$. Nor can $t$ be regular at the cusp, since an equality $t=P(t^3,t^4)/Q(t^3,t^4)$ with $Q(0)\ne 0$ would produce a nonzero degree-one term in $tQ$ and none in $P$. Thus pointwise bijectivity does not guarantee an algebraic inverse, even though the inverse is regular away from one singular point.

For incompressible simple shear $F=I+\gamma {\mathbf{e}}_1\otimes {\mathbf{e}}_2$ with Mooney-Rivlin energy $\alpha (I_1-3)+\beta (I_2-3)$, zero traction on the third-coordinate faces fixes the pressure reaction. The remaining Cauchy stress has $T_{11}=2\alpha {\gamma }^2$, $T_{22}=-2\beta {\gamma }^2$ and $T_{12}=2(\alpha +\beta )\gamma$. The material face $x_1=\text{constant}$ has current unit normal $\mathbf{n}=(1,-\gamma ,0)/\sqrt{1+{\gamma }^2}$ and tangent $\mathbf{m}=(\gamma ,1,0)/\sqrt{1+{\gamma }^2}$. Resolving $T\mathbf{n}$ gives $N=-2{\gamma }^2(\alpha +2\beta +\beta {\gamma }^2)/(1+{\gamma }^2)$ and $S=2\gamma (\alpha +\beta )/(1+{\gamma }^2)$. Since $N<0$, the traction-normal angle satisfies $\tan (\theta -\pi /2)=-N/|S|$, proving the displayed formula. Consequently its two-sided small-shear slope is $(\alpha +2\beta )/(\alpha +\beta )$. The reference normal cannot replace the transformed normal in this calculation.

For a left module $M$ over a unital associative algebra $A$, right multiplication in $A$ gives a right module structure on ${\mathrm{Hom}}_A(M,A)$ by $(f\cdot a)(m)=f(m)a$. This remains a module homomorphism because $f(bm)a=bf(m)a$; associativity and the unit verify the action axioms. When $A=M_d(\mathbb{C})$ and $M={\mathbb{C}}^d$ is its simple module of columns, every such map is $m\mapsto mr$ for a unique row $r$, making the Hom space the simple row right module. This illustrates how left and right simple modules correspond for a semisimple algebra.

The itch mite. Human-infecting forms burrow in the superficial skin.

A genus of mites associated with hair follicles and sebaceous structures.

This functional measures the surface area of the graph of a function. On a bounded domain it is finite for $u\in H^1$. Its directional derivative is ${\partial }_{v}F(u)=\int \nabla u\cdot \nabla v/\sqrt{1+|\nabla u{|}^2}$, by the dominated convergence theorem, since the pointwise difference quotients are bounded by $|\nabla v|$. The resulting dual-valued operator is monotone: the scalar integrand $p\mapsto \sqrt{1+|p{|}^2}$ is convex and its gradient is $p/\sqrt{1+|p{|}^2}$. It is not a nonlinear coercive operator on the full $H^1$ space, since it vanishes on constant functions whose $H^1$ norms can tend to infinity.

In the metric realization of a simplicial complex, the point on arm $n$ at parameter $t$ is $(1-t)e_v+t{e}_{u_n}$. Its distance from the common vertex is $\sqrt{2}t$. Consequently the vertex has a neighbourhood basis consisting of one uniformly short initial interval on every arm. Away from that vertex, each arm is locally isolated from the other arms.

The map $(n,t)\mapsto t{e}^{i/n}$, with all $t=0$ identified, is a homeomorphism ontoIt is a bijection, preserves the one-dimensional topology on each arm away from zero, and matches the uniform-radius neighbourhood bases at zero. The positive real limiting ray is omitted. The outer endpoints form an infinite closed discrete subset in the original metric realization, so the image is not compact. This planar fan and the weak realization of a countable star have different topologies at their common vertex.

A characteristic surface may remain smooth while its projection to the independent-variable plane fails to be locally invertible. At a generic fold, smooth changes of parameter and plane coordinates reduce this projection to $(u,v)\mapsto (u,v^2)$. There are two sheets on one side of the image of $v=0$, no sheet on the other, and one merged sheet on the fold itself. The image is a fold caustic. A characteristic graph can terminate there even though the space curves defining the surface continue smoothly. Vanishing of the Jacobian determinant detects a candidate singular locus; verifying the fold requires a nonzero quadratic term in its null direction, as in $v^2$.

For the first-order quasilinear partial differential equation $P(x,y,z)z_x+Q(x,y,z)z_y=R(x,y,z)$, the displayed ordinary differential equations trace curves tangent to every classical solution surface. Indeed, if $z=u(x,y)$ solves the equation and $x_s=P$, $y_s=Q$, then the chain rule gives $z_s=u_{x}P+u_{y}Q=R$. Conversely a surface swept out by these curves solves the equation wherever its projection to $(x,y)$ is invertible and hence represents a differentiable graph. The coefficients can depend on the unknown $z$, which distinguishes these equations from characteristic equations for a prescribed linear transport equation.

For a first-order partial differential equation with the displayed projected characteristic field, label the initial segment by $r=y(0)\in [0,c]$. The characteristic curves are $x=a-(a-b)e^s$, $y=r{e}^s$. They are the branches of straight lines through $(a,0)$ on the side $x<a$. The inverse coordinates areThey are unique exactly in the displayed sector. The Jacobian determinant is $-(a-b)e^{2s}\ne 0$, so the initial segment is noncharacteristic and its transported data determine a unique classical solution on the sector's interior whenever the characteristic equation for the dependent variable has a unique solution. The bounding rays inherit data from the two endpoints. The point $(a,0)$ is approached only at infinite parameter time, and characteristics on the opposite side cannot reach the initial segment.

A characteristic caustic is a locus where the coordinate map from initial-data and evolution parameters to physical coordinates has zero Jacobian determinant. The inverse function theorem then fails, and different characteristic branches can reach the same point. A classical solution constructed from initial data must be restricted to its appropriate invertible branch before such a fold. In geometrical optics the corresponding ray-envelope phenomenon is a caustic; in nonlinear first-order equations a fold can signal the onset of multiple gradients or shock formation by characteristic intersection.

For a fixed real unit vector $k$, the linear ordinary differential equation $\dot{r}=r-k\times r$ combines dilation with rotation about the axis $\mathbb{R}k$. Its solution from a line $s\ell$, with $|\ell |=1$, isSince $k\cdot r=s{e}^t\kappa$ and $|r|=|s|e^t$, the union of these characteristic curves is given by $|k\cdot r|=|\kappa ||r|$. For $0<|\kappa |<1$ it is a circular cone of semi-angle $\arccos |\kappa |$: varying $t$ supplies every azimuth and varying $s$ supplies every radius on both halves. For $\kappa =0$ it is a plane; for $|\kappa |=1$ it is only the axis. In the last case the line is itself a characteristic curve, so the initial data do not generate a two-dimensional surface. The cone is a smooth surface away from its vertex, and it is a graph of a function only on suitable patches. Invariance of a graph patch under characteristic curves holds only while the curve stays in its domain; global flow invariance requires continuation of that patch.